3 THE MODULUS FUNCTION ON A GROUP 257
a = 1 (because a > 0). This proves the first assertion. The second then follows
(13.14.3).
We deduce that if /is a locally /j-integrable function, then
(14.3.4.2) (/•AO~=(A-1/)-M.
(14.3.5) In particular, if G is unimodular> then for each ju-integrable function
the functions yC?)/, §(s)f and / are /i-integrable, and we have
.3.5.1) f f(sx) dtfd = (
(14.3
In particular, if A is any /i-integrable set in G, then
(14.3.5.2)
for all s e G.
When G is infinite and compact (resp. infinite and discrete) and therefore
unimodular (14.3.3), the normalized Haar measure on G is the unique Haar
measure \JL on G for which n(G) = 1 (resp. n({e}) = 1).
(14.3.6) Now let u be an automorphism of the topological group G. It is clear
that the image U~"I(IJL) of a left Haar measure \JL on G (13.1.6) is another left
Haar measure; hence (14.1.5) there exists a number a > 0, independent of the
choice of ju, such that U~I(IJL) = aju. This number a is called the modulus of the
automorphism uy and is denoted by modG(w) or mod(w). For every ^-integrable
function/we have therefore
f t f
(14.3.6.1) f(u~ \x)) d^(x) = (mod u) | /(x
and in particular, for any /i-integrable set A,
(14.3.6.2) Mw(A)) = (mod u
In particular, for each s e G, let is be the inner automorphism
Then we have /71 = &(S)Y(S\ an^ therefore
which proves that
(14.3J) mod(fs) =1 • /i) = (50A-1) • (8(^) = (A^^A-1) • (A^)^) = A"1 - u,